Finite Group Databasebeta

← #26139938

Subgroups — #26139938

Order 12032.

2222 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×Dic47⋊C4C211yesmaximal, characteristic
2C2×#15689008C212yesmaximal, complemented
3C2×#15688998C211yesmaximal, characteristic
4C2×C4×C22⋊Dic47C211yesmaximal, characteristic
5#20241643C212yesmaximal, complemented
6#20241640C212yesmaximal, complemented
7#20241623C212yesmaximal, complemented
8#20241610C211yesmaximal, characteristic
9#20241610C211yesmaximal, characteristic
10#20241610C211yesmaximal, characteristic
11C47×C22⋊#95C211yesmaximal, characteristic
12C24×Dic47C2211yescharacteristic
13C24×Dic47C2211yescharacteristic
14C22×C4⋊Dic47C2212yes—
15C22×Dic47⋊C4—28no—
16C22×Dic47⋊C4—24no—
17C22×Dic47⋊C4C2212yes—
18C22×Dic47⋊C4C2212yes—
19C22×C22⋊Dic47C2211yescharacteristic
20C22×C22⋊Dic47C2211yescharacteristic
21C22×C4×Dic47C2211yescharacteristic
22C22×C4×Dic47C2211yescharacteristic
23#15689008—28no—
24#15688998C418yescomplemented
25C4×C22⋊Dic47—24no—
26C2×C47⋊#95C2212yes—
27C2×C47⋊#95C2212yes—
28C2×C47⋊#95C2212yes—
29C2×C47⋊#95C2212yes—
30C2×C47⋊#95C2211yescharacteristic
31C2×C47⋊#95C2211yescharacteristic
32C2×C47⋊#95C2212yes—
33C2×C47⋊#95C2212yes—
34C2×C47⋊#95C2212yes—
35C24×C4×C47C2211yescharacteristic
36C22×C22⋊C4×C47C2211yescharacteristic
37C22×C22⋊C4×C47C2211yescharacteristic
38C2×#95×C47C2212yes—
39C2×#95×C47C2212yes—
40C23×Dic47—22no—
41C23×Dic47—22no—
42C23×Dic47—24no—
43C23×Dic47—24no—
44C23×Dic47—22no—
45C23×Dic47—24no—
46C23×Dic47—22no—
47C23×Dic47—22no—
48C23×Dic47—42no—
49C23×Dic47C2×C414yes—
50C23×Dic47C2311yescharacteristic
51C23×Dic47C2311yescharacteristic
52C23×Dic47C2312yes—
53C23×Dic47C2311yescharacteristic
54C23×Dic47C2311yescharacteristic
55C23×Dic47C2312yes—
56C2×C4⋊Dic47—24no—
57C2×C4⋊Dic47—24no—
58C2×C4⋊Dic47C2×C418yescomplemented
59C2×Dic47⋊C4—232no—
60C2×Dic47⋊C4—24no—
61C2×Dic47⋊C4—24no—
62C2×Dic47⋊C4—24no—
63C2×Dic47⋊C4—24no—
64C2×Dic47⋊C4—24no—
65C2×Dic47⋊C4D418yes—
66C2×C22⋊Dic47—22no—
67C2×C22⋊Dic47—22no—
68C2×C22⋊Dic47—22no—
69C2×C22⋊Dic47—22no—
70C2×C22⋊Dic47C2×C414yes—
71C2×C22⋊Dic47Q814yes—
72C2×C4×Dic47—22no—
73C2×C4×Dic47—22no—
74C2×C4×Dic47—24no—
75C2×C4×Dic47—42no—
76C47⋊#95—22no—
77C47⋊#95—22no—
78C47⋊#95—24no—
79C47⋊#95—24no—
80C47⋊#95—24no—
81C47⋊#95—24no—
82C47⋊#95—42no—
83C47⋊#95—42no—
84C47⋊#95C2×C418yescomplemented
85C25×C47C2311yescharacteristic
86C23×C4×C47—22no—
87C23×C4×C47—22no—
88C23×C4×C47—24no—
89C23×C4×C47—42no—
90C23×C4×C47D414yes—
91C23×C4×C47C2312yes—
92C23×C4×C47C2312yes—
93C23×C4×C47C2311yescharacteristic
94C23×C4×C47C2311yescharacteristic
95C2×C22⋊C4×C47—22no—
96C2×C22⋊C4×C47—22no—
97C2×C22⋊C4×C47—22no—
98C2×C22⋊C4×C47—22no—
99C2×C22⋊C4×C47—22no—
100C2×C22⋊C4×C47C2×C414yes—
101#95×C47—24no—
102#95×C47—42no—
103C24×C47—22no—
104C24×C47—22no—
105C24×C47—22no—
106C24×C47—22no—
107C24×C47C22×C412yes—
108C24×C47C2×D412yes—
109C24×C47C2×Q812yes—
110C24×C47#4112yes—
111C24×C47#4112yes—
112C24×C47#4112yes—
113C24×C47#4112yes—
114C24×C47C2411yescharacteristic
115C22×C4×C47—21no—
116C22×C4×C47—21no—
117C22×C4×C47—22no—
118C22×C4×C47—28no—
119C22×C4×C47—22no—
120C22×C4×C47—22no—
121C22×C4×C47—22no—
122C22×C4×C47—22no—
123C22×C4×C47—21no—
124C22×C4×C47—21no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—28no—
130C22×C4×C47—22no—
131C22×C4×C47—22no—
132C22×C4×C47—22no—
133C22×C4×C47—22no—
134C22×C4×C47—22no—
135C22×C4×C47—22no—
136C22×C4×C47—22no—
137C22×C4×C47—44no—
138C22×C4×C47—48no—
139C22×C4×C47—48no—
140C22×C4×C47C22×C414yes—
141C22×C4×C47C2×D412yes—
142C22×C4×C47C2×D412yes—
143C22×C4×C47C2×D414yes—
144C22×Dic47—28no—
145C22×Dic47—216no—
146C22×Dic47—28no—
147C22×Dic47—28no—
148C22×Dic47—22no—
149C22×Dic47—22no—
150C22×Dic47—22no—
151C22×Dic47—22no—
152C22×Dic47—216no—
153C22×Dic47—28no—
154C22×Dic47—22no—
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—21no—
164C22×Dic47—21no—
165C22×Dic47—21no—
166C22×Dic47—21no—
167C22×Dic47—22no—
168C22×Dic47—22no—
169C22×Dic47—22no—
170C22×Dic47—22no—
171C22×Dic47—22no—
172C22×Dic47—22no—
173C22×Dic47—22no—
174C22×Dic47—22no—
175C22×Dic47—22no—
176C22×Dic47—48no—
177C22×Dic47—48no—
178C22×Dic47—44no—
179C22×Dic47—44no—
180C22×Dic47C22×C412yes—
181C22×Dic47C22×C412yes—
182C22×Dic47C22×C414yes—
183C22×Dic47C2×D414yes—
184C22×Dic47C2×D414yes—
185C22×Dic47C2×Q812yes—
186C22×Dic47C2×Q812yes—
187C22×Dic47#4114yes—
188C22⋊C4×C47—28no—
189C22⋊C4×C47—44no—
190C22⋊Dic47—28no—
191C22⋊Dic47—28no—
192C4⋊Dic47—216no—
193Dic47⋊C4—216no—
194Dic47⋊C4—216no—
195C4×Dic47—44no—
196C4×Dic47—44no—
197C23×C47—24no—
198C23×C47—24no—
199C23×C47—22no—
200C23×C47—22no—
201C23×C47—24no—
202C23×C47—22no—
203C23×C47—22no—
204C23×C47—24no—
205C23×C47—24no—
206C23×C47—24no—
207C23×C47—24no—
208C23×C47—24no—
209C23×C47—24no—
210C23×C47—24no—
211C23×C47—24no—
212C23×C47—24no—
213C23×C47#11714yes—
214C23×C47C4×D414yes—
215C23×C47C4×Q814yes—
216C23×C47C22⋊Q814yes—
217C23×C47#12314yes—
218C23×C47#12514yes—
219C23×C47#12614yes—
220C23×C47C23×C411yescharacteristic
221C23×C47C22×D412yes—
222C23×C47C22×D411yescharacteristic
223C23×C47C22×Q811yescharacteristic
224C23×C47C2×#4112yes—
225C23×C47C2×#4111yescharacteristic
226C23×C47C2×#4111yescharacteristic
227C23×C47C2×#4111yescharacteristic
228C23×C47C2×#4111yescharacteristic
229C23×C47H(5,𝔽2)12yes—
230C23×C472−1+412yes—
231C2×C4×C47—22no—
232C2×C4×C47—22no—
233C2×C4×C47—24no—
234C2×C4×C47—24no—
235C2×C4×C47—24no—
236C2×C4×C47—24no—
237C2×C4×C47—24no—
238C2×C4×C47—22no—
239C2×C4×C47—22no—
240C2×C4×C47—22no—
241C2×C4×C47—416no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47—42no—
246C2×C4×C47—42no—
247C2×C4×C47—42no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—42no—
251C2×C4×C47—42no—
252C2×C4×C47—42no—
253C2×C4×C47—42no—
254C2×C4×C47—42no—
255C2×C4×C47#12014yes—
256C2×Dic47—232no—
257C2×Dic47—24no—
258C2×Dic47—24no—
259C2×Dic47—24no—
260C2×Dic47—24no—
261C2×Dic47—24no—
262C2×Dic47—24no—
263C2×Dic47—24no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—24no—
267C2×Dic47—24no—
268C2×Dic47—22no—
269C2×Dic47—22no—
270C2×Dic47—24no—
271C2×Dic47—24no—
272C2×Dic47—24no—
273C2×Dic47—24no—
274C2×Dic47—416no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—42no—
278C2×Dic47—42no—
279C2×Dic47—41no—
280C2×Dic47—41no—
281C2×Dic47—41no—
282C2×Dic47—41no—
283C2×Dic47—42no—
284C2×Dic47—41no—
285C2×Dic47—41no—
286C2×Dic47—42no—
287C2×Dic47—42no—
288C2×Dic47—42no—
289C2×Dic47—42no—
290C2×Dic47—42no—
291C2×Dic47C4×D418yes—
292#14039—471nomaximal, Sylow
293C22×C47—24no—
294C22×C47—24no—
295C22×C47—28no—
296C22×C47—28no—
297C22×C47—24no—
298C22×C47—24no—
299C22×C47—28no—
300C22×C47—28no—
301C22×C47—24no—
302C22×C47—24no—
303C22×C47#38718yescomplemented
304C22×C47C2×#11711yescharacteristic
305C22×C47C2×C4×D412yes—
306C22×C47C2×C4×D411yescharacteristic
307C22×C47C2×C4×Q811yescharacteristic
308C22×C47#52012yes—
309C22×C47C2×#12011yescharacteristic
310C22×C47C2×C22⋊Q811yescharacteristic
311C22×C47C2×#12311yescharacteristic
312C22×C47#52512yes—
313C22×C47#52511yescharacteristic
314C22×C47C2×#12511yescharacteristic
315C22×C47C2×#12611yescharacteristic
316C22×C47#53411yescharacteristic
317C22×C47#53611yescharacteristic
318C22×C47#53912yes—
319C22×C47#54612yes—
320C22×C47#54612yes—
321C22×C47#54812yes—
322C22×C47D4×Q812yes—
323C22×C47#55212yes—
324C22×C47#55212yes—
325C22×C47#55312yes—
326C22×C47#55412yes—
327C4×C47—24no—
328C4×C47—44no—
329C4×C47—42no—
330C4×C47—82no—
331Dic47—28no—
332Dic47—42no—
333Dic47—42no—
334Dic47—44no—
335Dic47—42no—
336Dic47—42no—
337C23×C4⋊C4—471no—
338C2×#387—472no—
339C2×#385—471no—
340C2×C4×C22⋊C4—471no—
341#1442—472no—
342#1441—472no—
343#1436—472no—
344C22⋊#95—471no—
345C22⋊#95—471no—
346C22⋊#95—471no—
347C22⋊#95D4711yescharacteristic
348C2×C47—28no—
349C2×C47C2×#38711yescharacteristic
350C2×C47C4×#12011yescharacteristic
351C2×C47#232211yescharacteristic
352C2×C47#235212yes—
353C2×C47#236012yes—
354C2×C47#240511yescharacteristic
355C2×C47#240911yescharacteristic
356C2×C47#247312yes—
357C2×C47#248112yes—
358C2×C47#248411yescharacteristic
359C2×C47#257911yescharacteristic
360C24×C4—471no—
361C24×C4—471no—
362C24×C4C2×D4711yescharacteristic
363C22×C4⋊C4—472no—
364C22×C4⋊C4—472no—
365C22×C4⋊C4—472no—
366C22×C4⋊C4—948no—
367C22×C4⋊C4—944no—
368C22×C22⋊C4—471no—
369C22×C22⋊C4—471no—
370C22×C22⋊C4C2×D4711yescharacteristic
371C22×C22⋊C4C2×D4711yescharacteristic
372C22×C42—471no—
373C22×C42—471no—
374#387—948no—
375#385—478no—
376C4×C22⋊C4—944no—
377C2×#95—472no—
378C2×#95—472no—
379C2×#95—472no—
380C2×#95—472no—
381C2×#95—472no—
382C2×#95—471no—
383C2×#95—471no—
384C2×#95—472no—
385C2×#95—472no—
386C2×#95C2×D4712yes—
387C2×#95C2×D4712yes—
388C47#1403911yescharacteristic, complemented, Sylow
389C25C22×D4711yescharacteristic
390C23×C4—22no—
391C23×C4—22no—
392C23×C4—24no—
393C23×C4—42no—
394C23×C4—472no—
395C23×C4—474no—
396C23×C4—471no—
397C23×C4—471no—
398C23×C4—471no—
399C23×C4—471no—
400C23×C4—472no—
401C23×C4—944no—
402C23×C4—944no—
403C23×C4—944no—
404C23×C4—942no—
405C23×C4—942no—
406C23×C4—942no—
407C23×C4—942no—
408C23×C4—942no—
409C23×C4—1882no—
410C23×C4D18814yes—
411C23×C4C22×D4712yes—
412C23×C4C22×D4712yes—
413C23×C4C22×D4711yescharacteristic
414C23×C4C22×D4711yescharacteristic
415C2×C4⋊C4—478no—
416C2×C4⋊C4—478no—
417C2×C4⋊C4—944no—
418C2×C4⋊C4—944no—
419C2×C4⋊C4—944no—
420C2×C4⋊C4—944no—
421C2×C4⋊C4—9432no—
422C2×C4⋊C4—944no—
423C2×C4⋊C4—944no—
424C2×C4⋊C4—944no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—474no—
431C2×C22⋊C4—474no—
432C2×C22⋊C4—942no—
433C2×C22⋊C4—942no—
434C2×C22⋊C4—942no—
435C2×C22⋊C4—942no—
436C2×C22⋊C4C4×D4714yes—
437C2×C42—942no—
438C2×C42—942no—
439C2×C42—944no—
440C2×C42—1882no—
441#95—24no—
442#95—42no—
443#95—478no—
444#95—944no—
445#95—944no—
446#95—944no—
447#95—944no—
448#95—942no—
449#95—942no—
450#95—1882no—
451#95—1882no—
452C24—22no—
453C24—22no—
454C24—22no—
455C24—22no—
456C24C2×C4×D4712yes—
457C24C2×D18812yes—
458C24#1062535012yes—
459C24#1062535212yes—
460C24#1062535212yes—
461C24#1062535212yes—
462C24Q8×D4712yes—
463C24C23×D4711yescharacteristic
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—28no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—28no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—474no—
490C22×C4—474no—
491C22×C4—474no—
492C22×C4—472no—
493C22×C4—472no—
494C22×C4—474no—
495C22×C4—472no—
496C22×C4—472no—
497C22×C4—9416no—
498C22×C4—948no—
499C22×C4—9416no—
500C22×C4—948no—
501C22×C4—948no—
502C22×C4—942no—
503C22×C4—942no—
504C22×C4—942no—
505C22×C4—942no—
506C22×C4—948no—
507C22×C4—942no—
508C22×C4—942no—
509C22×C4—942no—
510C22×C4—941no—
511C22×C4—941no—
512C22×C4—941no—
513C22×C4—941no—
514C22×C4—942no—
515C22×C4—942no—
516C22×C4—942no—
517C22×C4—941no—
518C22×C4—941no—
519C22×C4—942no—
520C22×C4—942no—
521C22×C4—941no—
522C22×C4—941no—
523C22×C4—942no—
524C22×C4—942no—
525C22×C4—942no—
526C22×C4—942no—
527C22×C4—942no—
528C22×C4—942no—
529C22×C4—1888no—
530C22×C4—1888no—
531C22×C4—1884no—
532C22×C4—1884no—
533C22×C4C2×C4×D4714yes—
534C22×C4C2×D18812yes—
535C22×C4C2×D18812yes—
536C22×C4D4×D4714yes—
537C4⋊C4—9416no—
538C4⋊C4—9416no—
539C4⋊C4—9416no—
540C22⋊C4—28no—
541C22⋊C4—44no—
542C22⋊C4—948no—
543C22⋊C4—948no—
544C42—1884no—
545C42—1884no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—24no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23#1309398014yes—
563C23C4×D18814yes—
564C23#1309398914yes—
565C23#1309400414yes—
566C23#1309400814yes—
567C23#1309401414yes—
568C23#1309401714yes—
569C23C22×C4×D4711yescharacteristic
570C23C2×D4×D4712yes—
571C23C22×D18811yescharacteristic
572C23C2×Q8×D4711yescharacteristic
573C23#41×D4712yes—
574C23C2×#1062535211yescharacteristic
575C23C2×#1062535211yescharacteristic
576C23C2×#1062535211yescharacteristic
577C23C2×#1062535011yescharacteristic
578C23#1309406912yes—
579C23#1309407112yes—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—416no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—478no—
605C2×C4—9432no—
606C2×C4—944no—
607C2×C4—944no—
608C2×C4—944no—
609C2×C4—944no—
610C2×C4—944no—
611C2×C4—944no—
612C2×C4—944no—
613C2×C4—944no—
614C2×C4—944no—
615C2×C4—944no—
616C2×C4—944no—
617C2×C4—942no—
618C2×C4—942no—
619C2×C4—944no—
620C2×C4—944no—
621C2×C4—942no—
622C2×C4—942no—
623C2×C4—18816no—
624C2×C4—1882no—
625C2×C4—1882no—
626C2×C4—1882no—
627C2×C4—1882no—
628C2×C4—1881no—
629C2×C4—1881no—
630C2×C4—1882no—
631C2×C4—1882no—
632C2×C4—1881no—
633C2×C4—1881no—
634C2×C4—1881no—
635C2×C4—1881no—
636C2×C4—1882no—
637C2×C4—1881no—
638C2×C4—1881no—
639C2×C4—1882no—
640C2×C4#1309399414yes—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—28no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—28no—
651C22#1568900818yescomplemented
652C22C2×#1309398011yescharacteristic
653C22C4×D4×D4712yes—
654C22C2×C4×D18811yescharacteristic
655C22C2×#1309398911yescharacteristic
656C22#1568960212yes—
657C22C2×#1309399411yescharacteristic
658C22C2×#1309400411yescharacteristic
659C22C2×#1309400811yescharacteristic
660C22#1568963212yes—
661C22#1568963511yescharacteristic
662C22C2×#1309401411yescharacteristic
663C22C2×#1309401711yescharacteristic
664C22#1568968511yescharacteristic
665C22#1568969911yescharacteristic
666C22#1568971512yes—
667C22#1568976012yes—
668C22#1568976012yes—
669C22#1568977912yes—
670C22#1568978612yes—
671C22#1568981012yes—
672C22#1568981012yes—
673C22#1568981612yes—
674C22#1568982712yes—
675C4—24no—
676C4—44no—
677C4—42no—
678C4—82no—
679C4—948no—
680C4—1882no—
681C4—1882no—
682C4—1882no—
683C4—1884no—
684C4—1882no—
685C2—28no—
686C2C2×#1568900811yescharacteristic
687C2C4×#1309399411yescharacteristic
688C2#2024605911yescharacteristic
689C2#2024631512yes—
690C2#2024638812yes—
691C2#2024671611yescharacteristic
692C2#2024675411yescharacteristic
693C2#2024753112yes—
694C2#2024765112yes—
695C2#2024767711yescharacteristic
696C2#2024873711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D47
commutatorC23×C47C23×C4
FittingC47×C22⋊#95C2
socle, cosocleC24×C47C24
prime radicalC25×C47C23
hypercenterC22⋊#95D47
hypocenterC47#14039
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#15689008C2semidirect
5. #20241643C2semidirect
6. #20241640C2semidirect
7. #20241623C2semidirect
24. #15688998C4semidirect
58. C2×C4⋊Dic47C2×C4semidirect
84. C47⋊#95C2×C4semidirect
303. C22×C47#387semidirect
388. C47#14039semidirect
651. C22#15689008semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×Dic47⋊C42
C2×#156890082
C2×#156889982
C2×C4×C22⋊Dic472
#202416432
#202416402
#202416232
#202416102
C47×C22⋊#952
#1403947